Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints
- Authors: Wu, Zhiyou , Jeyakumar, Vaithilingam , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 133, no. 1 (2007), p. 123-130
- Full Text: false
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- Description: We present sufficient conditions for the global optimality of bivalent nonconvex quadratic programs involving quadratic inequality constraints as well as equality constraints. By employing the Lagrangian function, we extend the global subdifferential approach, developed recently in Jeyakumar et al. (J. Glob. Optim., 2007, to appear; Math. Program. Ser. A, 2007, to appear) for studying bivalent quadratic programs without quadratic constraints, and derive global optimality conditions. © 2007 Springer Science+Business Media, LLC.
- Description: C1
A filled function method for nonlinear equations
- Authors: Wu, Zhiyou , Mammadov, Musa , Bai, Fusheng , Yang, Y. J.
- Date: 2007
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 189, no. 2 (2007), p. 1196-1204
- Full Text: false
- Reviewed:
- Description: In this paper, we propose a new global optimization approach based on the filled function method for solving box-constrained systems of nonlinear equations. The special properties of optimization problem are employed to construct a novel filled function. The objective function value can be reduced by half in each iteration of our filled function algorithm. Several numerical examples are presented to illustrate the efficiency of the present approach.
- Description: C1
- Description: 2003005618
A filled function method for constrained global optimization
- Authors: Wu, Zhiyou , Bai, Fusheng , Lee, Heung , Yang, Yongjian
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 39, no. 4 (2007), p. 495-507
- Full Text: false
- Reviewed:
- Description: In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems. © 2007 Springer Science+Business Media, Inc.
- Description: C1
- Description: 2003005513
A filled function method for box-constrained system of nonlinear equations
- Authors: Wu, Zhiyou , Mammadov, Musa , Bai, Fusheng
- Date: 2006
- Type: Text , Conference paper
- Relation: Paper presented at APCCAS 2006. IEEE Asia Pacific Conference on Circuits and Systems, Singapore : 4th -7th Dececmber, 2006 p. 623-626
- Full Text: false
- Reviewed:
- Description: In this paper, we present a global optimization method based on the filled function method to solve systems of nonlinear equations. Formulating a system of nonlinear equation into an equivalent global optimization problem, we manage to find a solution or an appropriate solution of the system of nonlinear equations by solving the formulated global optimization problem. A novel filled function method is proposed to solve the global optimization problem. Two numerical examples are presented to illustrate the efficiency of this method.
- Description: E1
- Description: 2003001840
Optimality conditions in global optimization and their applications
- Authors: Rubinov, Alex , Wu, Zhiyou
- Date: 2009
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 120, no. 1 SPEC. ISS. (2009), p. 101-123
- Full Text: false
- Reviewed:
- Description: In this paper we derive necessary and sufficient conditions for some problems of global minimization. Our approach is based on methods of abstract convexity: we use a representation of an upper semicontinuous function as the lower envelope of a family of convex functions. We discuss applications of conditions obtained to the examination of some tractable sufficient conditions for the global minimum and to the theory of inequalities. © 2007 Springer-Verlag.
A filled function method for constrained nonlinear integer programming
- Authors: Yang, Yongjian , Wu, Zhiyou , Bai, Fusheng
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Industrial and Management Optimization Vol. 4, no. 2 (May 2008), p. 353-362
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- Description: A filled function method is presented in this paper to solve constrained nonlinear integer programming problems. It is shown that for a given non-global local minimizer, a better local minimizer can be obtained by local search staring from an improved initial point which is obtained by locally solving a box-constrained integer programming problem. Several illustrative numerical examples are reported to show the efficiency of the present method.
- Description: C1
Global optimality conditions for mixed integer weakly concave programming problems
- Authors: Wu, Zhiyou , Quan, Jing , Bai, Fusheng
- Date: 2010
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 675-685
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- Description: In this paper, some necessary and some suffcient global optimality conditions for a class of mixed integer programming problems whose objective functions are the difference of quadratic functions and convex functions are established. The numerical examples are also presented to show the significance of the global optimality conditions for this class of programming problems. Copyright © 2010 Watam Press.
A filled function method for quadratic programs with binary constraints
- Authors: Wu, Zhiyou , Yang, Y. J. , Bai, Fusheng
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Vol. 58, no. 5 (2009), p. 585-598
- Full Text: false
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- Description: In this article, we first introduce a definition of a local minimizer for the quadratic programs with {-1, 1} constraints and show that the existing necessary condition of global optimality in some recent literature is in fact the necessary and sufficient condition of local optimality under the present definition. A filled function and the corresponding filled function algorithm then are proposed to get the global minimizer of the concerned minimization problem. The existing sufficient condition of global optimality is employed to form one of the termination rules of the algorithm. Several numerical examples are reported to show the efficiency of the present method for solving quadratic programs with binary constraints.
Quadratic smoothing approximation to 1/2-order exact penalty function
- Authors: Bai, Fusheng , Wu, Zhiyou
- Date: 2010
- Type: Text , Conference paper
- Relation: Paper presented at 1st International Conference on Green Circuits and Systems, ICGCS 2010, Shanghai : 21st-23rd June 2010 p. 409-413
- Full Text: false
- Description: In this paper, we propose a quadratic smoothing approximation to the 1/2-order exact penalty function. It is shown that when the penalty parameter of the smoothed penalty problem with the smoothing approximation function being penalty function is sufficiently large, any global minimizer of the smoothed penalty problem is an approximate feasible point of the original optimization problem, and the difference between the original objective function value on a global minimizer of the smoothed penalty problem and the global optimal value of the original problem can be controlled by the smoothing parameter which can be set in advance. Two numerical examples are reported to show the effectiveness of the proposed quadratic smoothing approximation method. © 2010 IEEE.
An auxiliary function method for constrained systems of nonlinear equations
- Authors: Wu, Zhiyou , Bai, Fusheng , Mammadov, Musa
- Date: 2008
- Type: Text , Conference paper
- Relation: Paper presented at 20th EURO Mini Conference: Continuous Optimization and Knowledge-Based Technologies, EurOPT-2008, Neringa, Lithuania : 20th-23rd May 2008 p. 259-265
- Full Text: false
- Description: In this paper, we propose an auxiliary function method to solve constrained systems of nonlinear equations. By introducing an auxiliary function, an unconstrained (box-constrained) optimization problem is constructed for a given constrained system of nonlinear equations. It is shown that any local minimizer of the constructed unconstrained optimization problem is an approximate solution to the given constrained system when parameters are appropriately chosen, and the precision for approximation can be preset. It is also shown that any accumulation point of the local minimizers of the constructed unconstrained optimization problems with a sequence of parameters tending to zero is a solution to the given constrained system of nonlinear equations.
A class of convexification and concavification methods for non-monotone optimization problems
- Authors: Wu, Zhiyou , Lee, Heung , Yang, Xin-Min
- Date: 2005
- Type: Text , Journal article
- Relation: Optimization Vol. 54, no. 6 (2005), p. 605-625
- Full Text: false
- Reviewed:
- Description: A class of convexification and concavification methods are proposed for solving some classes of non-monotone optimization problems. It is shown that some classes of non-monotone optimization problems can be converted into better structured optimization problems, such as, concave minimization problems, reverse convex programming problems, and canonical D.C. programming problems by the proposed convexification and concavification methods. The equivalence between the original problem and the converted better structured optimization problem is established.
- Description: C1
- Description: 2003003608
A recursive digital filter design using global optimization technique
- Authors: Wu, Zhiyou , Gu, Yanhong
- Date: 2006
- Type: Text , Conference paper
- Relation: Paper presented at APCCAS 2006, IEEE Asia Pacific Conference on Circuits and Systems, Singapore : 4th December, 2006
- Full Text: false
- Reviewed:
- Description: Close form analytical techniques for the design of a certain class of recursive digital filters such as the elliptic filter have appeared. Such close form analytical techniques are suitable for designing filters with piece-wise constant magnitude response. The design of recursive digital filters with arbitrary frequency response is a nonlinear optimization problem. Specifically, it belongs to the class of global bi-lever programming problem. Optimal solution for a global bi-lever programming problem is notoriously difficult to obtain. In this paper, the bi-lever programming problem is first converted into a differentiate one-lever problem. Consequently we not only prove that the global minimizer of the converted one-lever problem is an approximate global minimizer of the original bi-lever problem, but also a novel filled function method for the design of recursive digital filters meeting arbitrary frequency response specifications is proposed. Several design examples are presented to illustrate our new technique
- Description: E1
- Description: 2003001898
A novel monotonization transformation for some classes of global optimization problems
- Authors: Bai, Fusheng , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Asia-Pacific Journal of Operational Research Vol. 23, no. 3 (Sep 2006), p. 371-392
- Full Text: false
- Reviewed:
- Description: A novel monotonization method is proposed for converting a non-monotone programming problem into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavification methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.
- Description: C1
- Description: 2003003590
Global optimality conditions for some classes of optimization problems
- Authors: Wu, Zhiyou , Rubinov, Alex
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 145, no. 1 (2009), p. 164-185
- Full Text: false
- Reviewed:
- Description: We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity. © 2009 Springer Science+Business Media, LLC.
Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions
- Authors: Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 110, no. 3 (2007), p. 521-541
- Full Text: false
- Reviewed:
- Description: In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. © Springer-Verlag 2007.
- Description: C1
A new local and global optimization method for mixed integer quadratic programming problems
- Authors: Li, G. Q. , Wu, Zhiyou , Quan, Jing
- Date: 2010
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 217, no. 6 (2010), p. 2501-2512
- Full Text: false
- Reviewed:
- Description: In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable. Crown Copyright © 2010.
A dual criterion for maximal monotonicity of composition operators
- Authors: Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Set-Valued Analysis Vol. 15, no. 3 (2007), p. 265-273
- Full Text: false
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- Description: In this paper we present a dual criterion for the maximal monotonicity of the composition operator T:=A* SA, where S:Y→→ Y is a maximal monotone (set-valued) operator and A: X→ Y is a continuous linear map with the adjoint A*, X and Y are reflexive Banach spaces, and the product notation indicates composition. The dual criterion is expressed in terms of the closure condition involving the epigraph of the conjugate of Fitzpatrick function associated with S, and the operator A. As an easy application, a dual criterion for the maximality of the sum of two maximal monotone operators is also given. © 2006 Springer Science+Business Media B.V.
- Description: C1
An auxiliary function method for systems of nonlinear equations
- Authors: Wu, Zhiyou , Bai, Fusheng , Mammadov, Musa , Yang, Y. J.
- Date: 2007
- Type: Text , Conference paper
- Relation: Paper presented at 7th International Conference on Optimization: Techniques and Applications, ICOTA7, Kobe International Conference Center, Japan : 12th-15th December 2007
- Full Text: false
- Description: 2003005705
Global optimality conditions for mixed nonconvex quadratic programs
- Authors: Wu, Zhiyou , Bai, Fusheng
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Vol. 58, no. 1 (2009), p. 39-47
- Full Text: false
- Reviewed:
- Description: In this article, we present some global optimality conditions for mixed quadratic programming problems. Our approach is based on a L-subdifferential and an associated L-normal cone. Unlike most subdifferentials, the L-subdifferential is formed by functions that are not necessarily linear functions. We derive some sufficient and necessary global optimality conditions for mixed quadratic programs with box constraints and binary constraints.
Necessary and sufficient conditions for stable conjugate duality
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Nonlinear Analysis Vol. 64, no. 9 (2006), p. 1998-2005
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- Description: The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.
- Description: C1
- Description: 2003003596